Abstract : Recently the renormalization group predictions on the effect of disorder on pinning models have been put on mathematical grounds. The picture is particularly complete if the disorder is 'relevant' or 'irrelevant' in the Harris criterion sense: the question addressed is whether quenched disorder leads to a critical behavior which is different from the one observed in the pure, i.e. annealed, system. The Harris criterion prediction is based on the sign of the specific heat exponent of the pure system, but it yields no prediction in the case of vanishing exponent. This case is called 'marginal', and the physical literature is divided on what one should observe for marginal disorder, notably there is no agreement on whether a small amount of disorder leads or not to a difference between the critical point of the quenched system and the one for the pure system. In a previous work (arXiv:0811.0723) we have proven that the two critical points differ at marginality of at least exp(-c/beta^4), where c>0 and beta^2 is the disorder variance, for beta in (0,1) and Gaussian IID disorder. The purpose of this paper is to improve such a result: we establish in particular that the exp(-c/beta^4) lower bound on the shift can be replaced by exp(-c(b)/beta^b), c(b)>0 for b>2 (b=2 is the known upper bound and it is the result claimed in [Derrida, Hakim, Vannimenus, JSP 1992]), and we deal with very general distribution of the IID disorder variables. The proof relies on coarse graining estimates and on a fractional moment-change of measure argument based on multi-body potential modifications of the law of the disorder.